3.383 \(\int \frac{x^{3/2}}{\sqrt{b x^2+c x^4}} \, dx\)

Optimal. Leaf size=231 \[ \frac{\sqrt [4]{b} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{c^{3/4} \sqrt{b x^2+c x^4}}-\frac{2 \sqrt [4]{b} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{c^{3/4} \sqrt{b x^2+c x^4}}+\frac{2 x^{3/2} \left (b+c x^2\right )}{\sqrt{c} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}} \]

[Out]

(2*x^(3/2)*(b + c*x^2))/(Sqrt[c]*(Sqrt[b] + Sqrt[c]*x)*Sqrt[b*x^2 + c*x^4]) - (2
*b^(1/4)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*Ellip
ticE[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(c^(3/4)*Sqrt[b*x^2 + c*x^4]) +
(b^(1/4)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*Ellip
ticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(c^(3/4)*Sqrt[b*x^2 + c*x^4])

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Rubi [A]  time = 0.401544, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{\sqrt [4]{b} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{c^{3/4} \sqrt{b x^2+c x^4}}-\frac{2 \sqrt [4]{b} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{c^{3/4} \sqrt{b x^2+c x^4}}+\frac{2 x^{3/2} \left (b+c x^2\right )}{\sqrt{c} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)/Sqrt[b*x^2 + c*x^4],x]

[Out]

(2*x^(3/2)*(b + c*x^2))/(Sqrt[c]*(Sqrt[b] + Sqrt[c]*x)*Sqrt[b*x^2 + c*x^4]) - (2
*b^(1/4)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*Ellip
ticE[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(c^(3/4)*Sqrt[b*x^2 + c*x^4]) +
(b^(1/4)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*Ellip
ticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(c^(3/4)*Sqrt[b*x^2 + c*x^4])

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Rubi in Sympy [A]  time = 38.0441, size = 218, normalized size = 0.94 \[ - \frac{2 \sqrt [4]{b} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \sqrt{b x^{2} + c x^{4}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{c^{\frac{3}{4}} x \left (b + c x^{2}\right )} + \frac{\sqrt [4]{b} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{c^{\frac{3}{4}} x \left (b + c x^{2}\right )} + \frac{2 \sqrt{b x^{2} + c x^{4}}}{\sqrt{c} \sqrt{x} \left (\sqrt{b} + \sqrt{c} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)/(c*x**4+b*x**2)**(1/2),x)

[Out]

-2*b**(1/4)*sqrt((b + c*x**2)/(sqrt(b) + sqrt(c)*x)**2)*(sqrt(b) + sqrt(c)*x)*sq
rt(b*x**2 + c*x**4)*elliptic_e(2*atan(c**(1/4)*sqrt(x)/b**(1/4)), 1/2)/(c**(3/4)
*x*(b + c*x**2)) + b**(1/4)*sqrt((b + c*x**2)/(sqrt(b) + sqrt(c)*x)**2)*(sqrt(b)
+ sqrt(c)*x)*sqrt(b*x**2 + c*x**4)*elliptic_f(2*atan(c**(1/4)*sqrt(x)/b**(1/4))
, 1/2)/(c**(3/4)*x*(b + c*x**2)) + 2*sqrt(b*x**2 + c*x**4)/(sqrt(c)*sqrt(x)*(sqr
t(b) + sqrt(c)*x))

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Mathematica [C]  time = 0.0882347, size = 112, normalized size = 0.48 \[ \frac{2 i x^{5/2} \sqrt{\frac{c x^2}{b}+1} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}}\right )\right |-1\right )\right )}{\left (\frac{i \sqrt{c} x}{\sqrt{b}}\right )^{3/2} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(3/2)/Sqrt[b*x^2 + c*x^4],x]

[Out]

((2*I)*x^(5/2)*Sqrt[1 + (c*x^2)/b]*(EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c]*x)/Sqrt[
b]]], -1] - EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c]*x)/Sqrt[b]]], -1]))/(((I*Sqrt[c]
*x)/Sqrt[b])^(3/2)*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.018, size = 131, normalized size = 0.6 \[{\frac{b\sqrt{2}}{c}\sqrt{x}\sqrt{{1 \left ( cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}}\sqrt{{1 \left ( -cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-bc}}}}} \left ( 2\,{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) -{\it EllipticF} \left ( \sqrt{{1 \left ( cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}},{\frac{\sqrt{2}}{2}} \right ) \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)/(c*x^4+b*x^2)^(1/2),x)

[Out]

1/(c*x^4+b*x^2)^(1/2)*x^(1/2)*b/c*((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2)*2^(1/2
)*((-c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2)*(-x*c/(-b*c)^(1/2))^(1/2)*(2*Elliptic
E(((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2))-EllipticF(((c*x+(-b*c)^(1
/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{3}{2}}}{\sqrt{c x^{4} + b x^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3/2)/sqrt(c*x^4 + b*x^2),x, algorithm="maxima")

[Out]

integrate(x^(3/2)/sqrt(c*x^4 + b*x^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{\frac{3}{2}}}{\sqrt{c x^{4} + b x^{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3/2)/sqrt(c*x^4 + b*x^2),x, algorithm="fricas")

[Out]

integral(x^(3/2)/sqrt(c*x^4 + b*x^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{3}{2}}}{\sqrt{x^{2} \left (b + c x^{2}\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(3/2)/(c*x**4+b*x**2)**(1/2),x)

[Out]

Integral(x**(3/2)/sqrt(x**2*(b + c*x**2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{3}{2}}}{\sqrt{c x^{4} + b x^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3/2)/sqrt(c*x^4 + b*x^2),x, algorithm="giac")

[Out]

integrate(x^(3/2)/sqrt(c*x^4 + b*x^2), x)